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The Calculus Gallery: Masterpieces from Newton to Lebesgue
More than three centuries after its creation, calculus remains a dazzling intellectual achievement and the gateway into higher mathematics. This book charts its growth and development by sampling from the work of some of its foremost practitioners, beginning with Isaac Newton and Gottfried Wilhelm Leibniz in the late seventeenth century and continuing to Henri Lebesgue at the dawn of the twentieth--mathematicians whose achievements are comparable to those of Bach in music or Shakespeare in literature. William Dunham lucidly presents the definitions, theorems, and proofs. "Students of literature read Shakespeare; students of music listen to Bach," he writes. But this tradition of studying the major works of the "masters" is, if not wholly absent, certainly uncommon in mathematics. This book seeks to redress that situation.
Like a great museum, The Calculus Gallery is filled with masterpieces, among which are Bernoulli's early attack upon the harmonic series (1689), Euler's brilliant approximation of pi (1779), Cauchy's classic proof of the fundamental theorem of calculus (1823), Weierstrass's mind-boggling counterexample (1872), and Baire's original "category theorem" (1899). Collectively, these selections document the evolution of calculus from a powerful but logically chaotic subject into one whose foundations are thorough, rigorous, and unflinching--a story of genius triumphing over some of the toughest, most subtle problems imaginable.
Anyone who has studied and enjoyed calculus will discover in these pages the sheer excitement each mathematician must have felt when pushing into the unknown. In touring The Calculus Gallery , we can see how it all came to be.
Like a great museum, The Calculus Gallery is filled with masterpieces, among which are Bernoulli's early attack upon the harmonic series (1689), Euler's brilliant approximation of pi (1779), Cauchy's classic proof of the fundamental theorem of calculus (1823), Weierstrass's mind-boggling counterexample (1872), and Baire's original "category theorem" (1899). Collectively, these selections document the evolution of calculus from a powerful but logically chaotic subject into one whose foundations are thorough, rigorous, and unflinching--a story of genius triumphing over some of the toughest, most subtle problems imaginable.
Anyone who has studied and enjoyed calculus will discover in these pages the sheer excitement each mathematician must have felt when pushing into the unknown. In touring The Calculus Gallery , we can see how it all came to be.
256 pages, Hardcover
First published January 1, 2004
59 people are currently reading
605 people want to read
About the author
William Dunham
13 books83 followersAn American writer who was originally trained in topology but became interested in the history of mathematics and specializes in Leonhard Euler. He has received several awards for writing and teaching on this subject.
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Displaying 1 - 19 of 20 reviews
July 9, 2025
Newton was more interested in integration than differentiation and did most of his work using infinite series. (Infinite series had been discovered independently in India at least 150 years before, but I have yet to read that the Indians knew anything about calculus. China and the Islamic world were no longer major players in mathematics after the Mongol invasions.) Leibniz created most of the modern notation, including dy/dx for differentiation, the long s for integration and the Leibniz rule d(u v) = u dv + v du. The Fundamental Theorem of Calculus was in the air, but it didn't really say anything yet because there was no proper definition of the integral.
Calculus was originally treated as a branch of geometry, as dealing with the properties of curves. It was Cauchy who pointed out that differentiation, integration and infinite series are all examples of limits. He is therefore the father of Real Analysis as we now know it, although many of the questions he raised were not resolved until later.
The first truly analytical definition of integration was due to Riemann, and he also determined what kind of functions could be integrated. Others investigated what kind of functions could be differentiated, and showed that even when the derivative exists, it can't necessarily be integrated in the Riemann fashion.
Finally, Lebesgue defined a more general definition of integration and filled the last gaps in the Fundamental Theorem of Calculus by showing that where the derivative of a function exists, it can always be Lebesgue-integrated and always restores the original function.
So this book contains all the important ideas, but for that very reason it risks being a first year Real Analysis textbook.
Calculus was originally treated as a branch of geometry, as dealing with the properties of curves. It was Cauchy who pointed out that differentiation, integration and infinite series are all examples of limits. He is therefore the father of Real Analysis as we now know it, although many of the questions he raised were not resolved until later.
The first truly analytical definition of integration was due to Riemann, and he also determined what kind of functions could be integrated. Others investigated what kind of functions could be differentiated, and showed that even when the derivative exists, it can't necessarily be integrated in the Riemann fashion.
Finally, Lebesgue defined a more general definition of integration and filled the last gaps in the Fundamental Theorem of Calculus by showing that where the derivative of a function exists, it can always be Lebesgue-integrated and always restores the original function.
So this book contains all the important ideas, but for that very reason it risks being a first year Real Analysis textbook.
April 2, 2020
It’s a quite interesting book on the history of Mathematics but you need to have a certain expertise on the subject to read it. I loved the writing style, it’s like you’re reading an epic story about sports tails. When a mathematician delivers a proof, the author makes it seem like he’s writing about a Tom Brady game winning drive or like Diego Maradona put Argentina on his back, except that instead of Brady and Maradona it’s Euler and Newton putting calculus on their backs. A rare ode to a past of glorious Math.
June 14, 2020
This is one of the most interesting books on the history of Calculus that I have ever read. It does require a moderate amount of mathematical knowledge (although not more than the standard first year undergraduate Analysis courses), but it is written with such a brilliance that one reads it with the eagerness more frequently experienced when reading a good thriller. But then, the history of Mathematical Analysis is, when we look at it in the proper way, one of the most fascinating and thrilling episodes in the intellectual history of mankind. This book is but one of the different stories that can be written: not being the history of Calculus, not even a history, it is, as the title indicates, a gallery, like an art gallery: reading along it we travel from the founding fathers Newton and Leibitz, until the pinacle of rigor and generality (and beauty!!) attained in the beginning of the 20th Century by Baire and Lebesgue. Along the way we visit some of the brilliant ideas of the Bernoulli brothers, Euler, Cauchy, Riemann, Liouville, Weierstrass, Cantor, and Volterra, and we see how, in two and a half centuries, the combined work of these (and others) outstanding minds shaped one of the most beautiful and powerful of all human creations. Like in any art gallery, a lot of names, some of then genius, are missing, but what is there is enough to tell a story, to disquiet and to awe the visitor. All in all, this is a magnificent book that all teachers and students of mathematics should read. It is also a work that should sadden us for the beauty herein is not likely to be appreciated by many more. It comes to mind the following famous poem by Fernando Pessoa, one of the most celebrated of all Portuguese poets (in my loose translation): Newton's binomial is as beautiful as the Venus of Milo. The trouble is that few people can be aware of this. And the (generalized) Newton's binomial expansion is just the beginning: it is the very first section of the first chapter in this book...
August 3, 2020
This is one of the most brilliant book about the history of real analysis , starting from the founding fathers of calculus like Newton, Leibniz and then exploring the ideas of the brothers Bernoulli, Euler and of course the genius and exactitude of Cauchy and Riemann and ending with the work of Bair and Lesbegue who omitted all confusion on the subject and relaunch calculus on much solid ground.All in all, reading this book is like reading a thriller although it needs some basic knowledge of mathematics at least a first year undergraduate level to understand the variety of the brilliant logical proofs present along this book. This book is not a text book neither a work of literature destined to ley man but it is something in between which makes it really enjoyable.
October 16, 2020
I am an experienced high school calculus teacher and I found this book to be quite demanding. It is not for the general reader. Dunham''s Journey Through Genius and The Mathematical Universe are much more accessible for the general audience.
April 10, 2022
Extremely well written. Recommend to people who have studied analysis at University for without this the latter chapters are rather turgid.
May 13, 2020
This is quite possibly the best book on mathematics I have ever read. Wonderfully written, both for the technical details as well as the colorful analogies and historical tidbits. With every chapter, I learned something new, gained fresh insight, and felt ever more interested in the ongoing tale.
August 31, 2020
One of my all time favourite books
The history of calculus from newton to weinstrauss.
It takes you through all the theorems you learn in analysis at university, but sets them in their historic context
The history of calculus from newton to weinstrauss.
It takes you through all the theorems you learn in analysis at university, but sets them in their historic context
September 11, 2020
Excellent reading for understand the history of analysis
February 24, 2024
informative inspiring reflective medium-paced
4.25
4.25
March 3, 2015
This book is a wonderful hybrid of mathematics and "art", hence the title, "The Calculus Gallery". It is nice to be able to walk though the halls of higher mathematics and appreciate some of it's major "works" just as one would paintings on the wall of a great art gallery. It brings mathematics itself into the "great conversation" of art, literature, and music as Raphael depicts in his famous fresco, "The School of Athens". Here we see geometers working in a great hall along with scribes and musicians and poets with Aristotle and Plato walking astride each other, engaged in conversation. Indeed, everyone is talking because at its core scholarship is about conversation and the love of serious books, is it not? This is the kind of experience I craved when I first came to the university! If I had experienced it perhaps I wouldn't have indulged in the cruder pleasures and diversions that came my way.
Which leads me to the point that it is also disheartening because nowadays we rarely expect such things from the academy because its members are busy carving out professional reputations through the deployment of high technical skill. To the extent that it is present conversation is primarily a means to determine one's rank and, rather than books, the primary source of information is the professional journal. Under this model, students are simply to be programmed into their fields of choice and according to their raw talents--wisdom and refinement of the soul are completely omitted. Per Russell Jacoby's book "The Last Intellectuals", this has contributed to the schism between academia and the educated public and the decline of what he called public intellectuals. The new intellectuals even disdain laymen because they are not perceived capable of following the delecate nuances of their, often counterintuitive, reasoning.
On the positive side, it occurs to me that the gallery approach could be put to beneficial effect teaching mathematics; compared to the current and often arid, pre-digested, textbook oriented approaches that seem to dominate university level teaching. (That was my experience in undergraduate and graduate school, anyway, so I'll defer to those who have been through university more recently to say whether or not it is still the case. Love to hear from you, perhaps we could start a discussion?)
Which leads me to the point that it is also disheartening because nowadays we rarely expect such things from the academy because its members are busy carving out professional reputations through the deployment of high technical skill. To the extent that it is present conversation is primarily a means to determine one's rank and, rather than books, the primary source of information is the professional journal. Under this model, students are simply to be programmed into their fields of choice and according to their raw talents--wisdom and refinement of the soul are completely omitted. Per Russell Jacoby's book "The Last Intellectuals", this has contributed to the schism between academia and the educated public and the decline of what he called public intellectuals. The new intellectuals even disdain laymen because they are not perceived capable of following the delecate nuances of their, often counterintuitive, reasoning.
On the positive side, it occurs to me that the gallery approach could be put to beneficial effect teaching mathematics; compared to the current and often arid, pre-digested, textbook oriented approaches that seem to dominate university level teaching. (That was my experience in undergraduate and graduate school, anyway, so I'll defer to those who have been through university more recently to say whether or not it is still the case. Love to hear from you, perhaps we could start a discussion?)
March 29, 2016
Calculus, by and large, has been considered one of the most difficult subjects undergraduate students have ever dealt with for decades. Just because most calculus textbooks are full of mathematical concepts, theorems and tools without any in-depth explanation of their foundations and the way they have elvoved onto something we know up to date. As if everything was just coming out of the mathematical "black hole".
William Dunham, the author, really did a great job to cover all of the most remarkable foundations of calculus, from the initial concepts as Newton, Leibniz had created to the dilemmas of "ghost of the departed quantities" Berkeley once critised. Then we know how Cauchy, Weierstrass and many others came to the stage to fine-tune those which have played a major roles in modern mathematical analysis.
Then he introduces the journey of questioning the nature of real numbers from which mathematicans like Cantor, Baire made great efforts laying the foundations of mathematical concepts and reasonings.
I would strongly recommend this book to every undergrad students before jumping into calculus at college to demystify any obstacles that may arise along the way.
William Dunham, the author, really did a great job to cover all of the most remarkable foundations of calculus, from the initial concepts as Newton, Leibniz had created to the dilemmas of "ghost of the departed quantities" Berkeley once critised. Then we know how Cauchy, Weierstrass and many others came to the stage to fine-tune those which have played a major roles in modern mathematical analysis.
Then he introduces the journey of questioning the nature of real numbers from which mathematicans like Cantor, Baire made great efforts laying the foundations of mathematical concepts and reasonings.
I would strongly recommend this book to every undergrad students before jumping into calculus at college to demystify any obstacles that may arise along the way.
February 11, 2016
When I studied calculus I always longed for a bit of historical context of its development because I like to place the things I learn into an historical perspective.
Finally, my curiosity got pleased as William Dunham's book does exactly that: He presents a timeline in which the most important theorems and definitions of calculus got developed, giving you a sense of how these great mathematicians build steadily on the ideas presented by their precursors.
"The Calculus Gallery" is not a calculus textbook in the strict sense, so you should bring enough basic knowledge of the matter with you - but Dunham will help you to appreciate these things you already learned in a (most probably) dry calculus course by showing how creative thinkers wrestled with the subject and thereby developed some of the most important intellectual achievments.
Finally, my curiosity got pleased as William Dunham's book does exactly that: He presents a timeline in which the most important theorems and definitions of calculus got developed, giving you a sense of how these great mathematicians build steadily on the ideas presented by their precursors.
"The Calculus Gallery" is not a calculus textbook in the strict sense, so you should bring enough basic knowledge of the matter with you - but Dunham will help you to appreciate these things you already learned in a (most probably) dry calculus course by showing how creative thinkers wrestled with the subject and thereby developed some of the most important intellectual achievments.
September 7, 2016
Math for people who like stories!
When I took an Analysis course I hated it, but was surprised by the very different picture of Calculus that it provided.
This book presents Analysis as a riveting series of developments from the 17th to 20th century. Each chapter focuses on a specific mathematician and a few of their most surprising results. The proofs are concise and act in service of the big picture, which is a roller coaster of funky observations, counterexamples, and generalizations.
It's a great look at a body of knowledge starting with Newton/Leibniz and ending with Lebesgue, who took the intuitive Riemann integral and (literally) turned it sideways, solving many open questions in the process.
When I took an Analysis course I hated it, but was surprised by the very different picture of Calculus that it provided.
This book presents Analysis as a riveting series of developments from the 17th to 20th century. Each chapter focuses on a specific mathematician and a few of their most surprising results. The proofs are concise and act in service of the big picture, which is a roller coaster of funky observations, counterexamples, and generalizations.
It's a great look at a body of knowledge starting with Newton/Leibniz and ending with Lebesgue, who took the intuitive Riemann integral and (literally) turned it sideways, solving many open questions in the process.
April 6, 2012
An okay book, not as good as expected
August 6, 2014
Dunham's "Journey Through Genius" is more general, accessible and broadly appealing, so start with that. If you loved it, read this next.
December 29, 2013
An essential read for any undergrad taking real analysis.
January 24, 2014
Difficult, yet rewarding
October 22, 2015
Absolutely brilliant and thoroughly engaging tour of a wonderful subject!
Displaying 1 - 19 of 20 reviews